# Phase 2: IV Estimation — Interpretation Notes

## Summary

IV estimation with lagged fertility instruments produces three key findings:

1. **First stages are strong** (F > 10-50 for most models), confirming that lagged demographics
   are powerful predictors of current demographics. The Bartik instrument is even stronger (F > 400).

2. **Second-stage results are highly unstable across instrument choices**: the Z₁ coefficient
   ranges from -2113 to +607 depending on lag length and instrument set. This instability
   suggests the exclusion restriction may be violated — lagged fertility affects CA through
   channels other than current age structure (e.g., institutional development, path-dependent
   policy choices, or persistent economic structure).

3. **Hausman test fails to reject OLS consistency** (p=0.41 full sample, p=0.76 ex-CCA),
   meaning there is no detectable endogeneity bias in the OLS estimates. The OLS coefficients
   are our best estimates.

## Detailed findings

### Overidentification failure (Model G)
- Hansen J = 582.2 (p=0.000) with 17 lagged age shares as instruments
- This means at least some lagged age shares affect CA independently of current Z
- Most likely channel: lagged fertility → institutional development → CA
- Or: lagged fertility → persistent economic structure → CA

### Instability across lag lengths
- 20yr lag: Z₁ = +226 (sig at 5%)
- 25yr lag: Z₁ = -66 (insignificant)
- 30yr lag: Z₁ = +607 (insignificant, SEs explode)
- This pattern suggests the instrument is picking up different variation at different lags,
  not a stable causal effect

### Bartik instrument (Model H)
- Extremely strong first stage (F > 400) but Z coefficients 8× OLS
- Likely violation: global demographic trends → global r* → individual CA
- The global shift component is NOT exogenous to individual CAs because it operates
  through the world interest rate (general equilibrium channel)

### Anderson-Rubin results
- AR tests reject H0 across all models (p < 0.03), including ex-CCA (p=0.000)
- AR is robust to weak instruments — says demographics matter "somehow"
- But unstable 2SLS coefficients mean we can't reliably estimate the magnitude
- The AR result for ex-CCA is the most interesting: demographics affect CA
  even without CCA, but the effect is imprecisely estimated

## Implications for the paper

1. **IV is not the path to causal identification here**. The fundamental problem is that
   demographics are slow-moving and correlated with institutional development — standard
   IV cannot cleanly separate these channels.

2. **The Hausman non-rejection is good news**: it means OLS is not detectably biased,
   so the baseline OLS results are defensible as correlation-based estimates.

3. **The staggered DiD approach (Phase 3) is more promising** because it exploits
   within-country, within-time variation in capital account opening — a discrete policy
   event — rather than trying to instrument demographics.

4. **The Bartik result is informative about mechanism**: the fact that global demographic
   trends predict individual CAs (even through a potentially contaminated channel)
   suggests demographics operate at least partly through global capital market clearing.

## What to report in the paper

- Full first-stage diagnostics (strong instruments)
- Anderson-Rubin tests (robust joint significance)
- Hausman test (no detectable endogeneity)
- Overidentification failure (exclusion restriction concern)
- Instability across instrument choices (honest reporting)
- Conclusion: IV cannot isolate a clean causal effect due to slow-moving nature of demographics,
  but non-rejection of Hausman means OLS estimates are not detectably biased
